Abstract

We introduce a high performance parallelization to the PSTD solution of Maxwell equations by employing the fast Fourier transform on local Fourier basis. Meanwhile a reformatted derivative operator allows the adoption of a staggered-grid such as the Yee lattice in PSTD, which can overcome the numerical errors in a collocated-grid when spatial discontinuities are present. The accuracy and capability of our method are confirmed by two analytical models. In two applications to surface tissue optics, an ultra wide coherent backscattering cone from the surface layer is found, and the penetration depth of polarization gating identified. Our development prepares a tool for investigating the optical properties of surface tissue structures.

Figures (10)

(a). The curve of function f(x) stacked on top of a schematic diagram illustrating overlapping domain decomposition and the bell function; and (b) the first derivative of f(x) given by Eq. (4) vs. analytical results. In (a), the x axis is divided by the vertical dashed lines into multiple subdomains, which are in turn mapped into different computation nodes. Each node obtains from its neighbors a copy of data in the thin neighborhood just outside its boundaries, referred to as the overlapping region; meanwhile the subdomain itself forms the non-overlapping region. The data in the overlapping and non-overlapping regions are then multiplied by the bell function to produce the local data for the local FFT. The bell function, as depicted by the blue curve, has a flat top 1 and gradually decreases to 0 at both the left and the right ends, and imposes the periodic condition on the local data as required by FFT. The derivatives concatenated from all the non-overlapping regions reproduce the derivatives in the whole computation region. Here for illustrative purposes, the width of the subdomains in (a) is not proportional to what we employed in actual calculations. In (b), note there are actually two curves in the plot, but they agree so well that they essentially coincide. Because Eq. (1) has an analytical expression, its exact first derivative can be strictly calculated. So the discrete results in (b) demonstrate the remarkable accuracy of Eq. (4) and FFT with local Fourier basis. Here for clarity, only region [-50,50] is shown in (b). Situations in other regions are similar.

Differential scattering cross section . dσ/dθ. as predicted by SLPSTD vs. the exact Mie solution, in linear scale (left axis) and logarithmic scale (right axis). Gird resolution is set to 0.098 μm, corresponding to 5 samplings per wavelength in polystyrene. The two curves coincide when plotted on linear scale. Their discrepancy is only visible on the logarithmic scale at the order of 10−6.

Radiation intensities from an electric dipole embedded in a concentric dielectric sphere, as predicted by SLPSTD and the analytical solution. As shown in the inset, the concentric sphere simulates a biological cell and is composed of two layers. The inner core and the outer shell have the diameters and refractive indices of cell nucleus and cytoplasm, respectively. The surrounding medium is water. A harmonic electric dipole off-centered along the z axis simulates a Raman emitter or fluorophore. Origin of the coordinates is at the sphere center and θ is the polar angle.

Snapshots of the electric field Ex on the plane z=1.4 μm, (a) parallel solution by SLPSTD and (b) sequential solution by collocated-grid PSTD with global Fourier basis. In (b), the actual signal is completely overwhelmed by spurious artifacts.

Schematic diagram of the setup for EBS simulation. The tissue-like medium is a thin rectangular suspension of polystyrene beads, with volume equal to 100×100×50 μm3. Note the medium itself is immerged in water, not air. Microspheres of 2-μm diameter are uniformly and randomly positioned in the rectangle at 2.9% vol. concentration. A plane wave linearly polarized in the incidence plane is delivered at 15° from normal to avoid the specular reflection. Backscattering angle θ is defined as the angular deviation from the reverse of incidence. Scattered outgoing wave is absorbed by setting perfectly matched layers in the PSTD program. To suppress speckles, results are averaged over 21 different frequencies centered at f0=3.82×1014Hz.

The calculated EBS cone using the same parameters in Kim’s experiment [6], averaged over 4 sets of scattering medium realizations. The origin of the slow descending slop in the tail of the curve (0.9°<θ<3°) is still unknown.

Simulation results for a series of τ: (a) the total signal Itot(θ)and (b) the difference signal Idif(θ). I0 is the intensity of the incident light. Peaks at θ≤0.170 contain specular reflection components.